The McCarthy–Osburn–Straub conjecture for two cellular integrals with N = 7: a Frobenius structure in two summation variables

For every N ≥ 5, Brown's cellular integrals on the moduli spaces M_{0,N} produce integer sequences A_σ(n). For N = 5 and N = 6 they are the sequences of Apéry's irrationality proofs of ζ(2) and ζ(3), which satisfy the supercongruences A_σ(mp^r) ≡ A_σ(mp^(r−1)) (mod p^(3r)) for every prime p ≥ 5 (Coster). McCarthy, Osburn and Straub conjectured (Conjecture 1.3 of their 2020 paper) that these supercongruences hold for every N ≥ 5. To the author's knowledge, no case beyond N = 6 was known. For N = 7 there are five distinct sequences, attached to fourth-order Calabi–Yau operators; one of them is a(n)², for which the conjecture follows from Coster's theorem. This paper proves the conjecture for two of the cellular integrals with N = 7, those attached to the Calabi–Yau operators #193 and #198 of Almkvist, van Enckevort, van Straten and Zudilin, for which no hypergeometric or modular description is known. The same results prove both supercongruence conjectures of Bala for these two sequences (OEIS A376460 and A376461), along mp^r and along mp^r − 1. The two sequences come from a single double sum, F(y) = Σ_{k,l} C(y,k)² C(y,l)² C(k+l,k) C(y+k+l,k+l). Its values at y ≥ 0 give u(n) = F(n) = 1, 11, 559, 42923, … (A376460), and its values at the negative integers give v(n) = (−1)^n F(−n−1) = 1, 7, 199, 8359, … (A376461). Main results. Theorem A. For every prime p ≥ 5 and every integer m ≠ 0, F(mp^r) = Φ_m(p^r) for all r ≥ 0, where Φ_m is a power series that converges on a disc of radius larger than 1. Among series converging on the unit disc it is unique. Its coefficients are p-adic integers, and Φ_m(z) − Φ_m(0) lies in p³ z³ Z_p[[z]]. Theorem B. For every prime p ≥ 5 and every integer m prime to p, F(mp^r − 1) = Φ̂_m(p^r) for all r ≥ 0, with a power series Φ̂_m that has the same properties. Corollary C. F(np^r) ≡ F(np^(r−1)) and F(np^r − 1) ≡ F(np^(r−1) − 1) (mod p^(3r)) for every prime p ≥ 5, every integer n ≠ 0 and every r ≥ 1. If p does not divide n, the normalized differences (F(np^r) − F(np^(r−1)))/p^(3r) are congruent modulo p^(r−1) to c_3(n)(p³ − 1)/p³, where c_3(n)/p³ is a p-adic integer that does not depend on r; the same holds along np^r − 1. Corollary D. For every prime p ≥ 5 and all integers n, r ≥ 1, u(np^r) ≡ u(np^(r−1)), u(np^r − 1) ≡ u(np^(r−1) − 1), v(np^r) ≡ v(np^(r−1)) and v(np^r − 1) ≡ v(np^(r−1) − 1) (mod p^(3r)). In particular Conjecture 1.3 of McCarthy, Osburn and Straub holds for the binomial sums attached to σ = (7,2,4,1,6,3,5) and σ = (7,2,5,1,4,6,3), and both supercongruence conjectures of Bala hold for A376460 and A376461. Theorem E (structure). There is a 5×5 matrix Θ(y) of p-adic power series with V(py) = Θ(y)V(y) for every integer y, where V(y) is the vector of the five double moments of the summand with weights 1, k, l, kl, l²; a second matrix Θ̂ does the same along the shifted tower, V(py − 1) = Θ̂(y)V(y − 1). The module of moments has dimension exactly 5 at every level, and the reduction to its basis is polynomial in the level. Θ(y) ≡ diag(1, p, p, p², 0) + p²(2y E_42 − y² E_40) (mod p³). Θ(0) has eigenvalues 1, p, p, p², p⁴, and det Θ(y) = p⁸. The vectors (0, 1, 0, y, 0) and (0, 0, 1, y, y(y+1)(y+2)) are exact eigen-sections. Θ commutes with an explicit shift matrix R(x) of determinant −1 that realizes the recurrence in the upper index. Method. (1) Weights at every level. Writing k and l with their usual p-adic digits, the quotient of the summand at levels py and y is a power series in all three variables with no poles. Because it is analytic in y, the factorization extends to every y in Z_p, in particular to the negative integers, which carry the second sequence. (2) A module without exceptional levels. The module of moments has dimension exactly 5 for every level, and the pivots of the reduction do not depend on the level. (3) The eigenvalue p⁴. The eigenvalues 1, p, p, p² come from jets at the origin. The fifth one, p⁴, comes from the interior of the summation lattice. It is detected by a functional on the squares [0, N−1]², suggested by the limit T(y,k,l)/y⁴ → C(k+l,k)/(k²l²) as y → 0. (4) Resonances paired with a cone. The spectrum has nine resonances. The exact sections and jets on the diagonal dispose of seven of them. The other two are removed by pairing with an approximate section from the left that comes from the cone (y−k)²(y−l)², which is stable under Frobenius. (5) The shift matrix. A commutation theorem between Frobenius and the shift matrix, proved with twisted modules of moments, gives det Θ = p⁸, the exact eigen-sections, and the link between the towers mp^r and mp^r − 1. (6) Integrality with an eigenvalue of valuation 4. The integrality criterion of the earlier papers needs all eigenvalues of valuation at most 3. The missing digit is recovered from the last row of Θ, with the same cone. The eigen-sections then have p-adic integral coefficients and form a unimodular matrix. (7) The second sequence for free. At the negative levels the summand has finite support and the Frobenius identity still holds, so the second sequence follows from the Frobenius structure of the first. Verification. Apart from five steps, the proofs do not depend on computer calculations. Those steps are finite computations with explicit polynomials and rational functions, listed in Appendix B of the paper: (B1) a bound for the valuations of the Frobenius coefficients; (B2) the reduction of the cone (y−k)²(y−l)² in the module of moments; (B3) the shift matrix and its determinant; (B4) the reduction in the twisted modules; (B5) two telescoping certificates that identify F with Bala's definitions. All five are reproduced in exact rational arithmetic by the scripts in this record. Independently of the proofs, the scripts also check Corollaries C and D numerically, computed directly from F, and the identification with the binomial sums of McCarthy, Osburn and Straub and with A376460 and A376461. Authorship. The author of this work is Roberto Blanco Gómez. The structural tests behind this work were carried out with SyntheticMind, the mathematical engine that the author created, designed and programmed himself in Rust. The AI system Claude (Anthropic) collaborated as an assistant, under the author's direction, in developing and checking the arguments, in independent review passes over the proofs, in the verification scripts and in drafting, translating and editing the text. The author has checked every statement and every proof and takes full responsibility for the content. Files: Blanco_Gomez_2026_MOS_N7_cellular_integrals_EN.pdf: the paper. Blanco_Gomez_2026_MOS_N7_cellular_integrals_ES.pdf: the Spanish version, with the same content. verification.zip: the verification scripts (Python 3 with SymPy), README.md, run_all.py, the certificates and the complete output. sources.zip: LaTeX sources of both versions.